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Properties

Label 2-30e2-100.23-c1-0-44
Degree 22
Conductor 900900
Sign 0.257+0.966i0.257 + 0.966i
Analytic cond. 7.186537.18653
Root an. cond. 2.680772.68077
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 0.574i)2-s + (1.34 − 1.48i)4-s + (−0.0819 + 2.23i)5-s + (1.03 + 1.03i)7-s + (−0.880 + 2.68i)8-s + (−1.17 − 2.93i)10-s + (−3.35 − 4.61i)11-s + (−0.392 − 2.48i)13-s + (−1.92 − 0.742i)14-s + (−0.405 − 3.97i)16-s + (−4.26 − 2.17i)17-s + (−1.72 − 5.31i)19-s + (3.20 + 3.11i)20-s + (6.98 + 4.04i)22-s + (−0.0446 + 0.281i)23-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.670 − 0.742i)4-s + (−0.0366 + 0.999i)5-s + (0.390 + 0.390i)7-s + (−0.311 + 0.950i)8-s + (−0.372 − 0.928i)10-s + (−1.01 − 1.39i)11-s + (−0.108 − 0.687i)13-s + (−0.515 − 0.198i)14-s + (−0.101 − 0.994i)16-s + (−1.03 − 0.526i)17-s + (−0.396 − 1.21i)19-s + (0.717 + 0.697i)20-s + (1.49 + 0.861i)22-s + (−0.00931 + 0.0587i)23-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.257+0.966i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+1/2) L(s)=((0.257+0.966i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.257+0.966i0.257 + 0.966i
Analytic conductor: 7.186537.18653
Root analytic conductor: 2.680772.68077
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ900(523,⋅)\chi_{900} (523, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :1/2), 0.257+0.966i)(2,\ 900,\ (\ :1/2),\ 0.257 + 0.966i)

Particular Values

L(1)L(1) ≈\approx 0.433980−0.333562i0.433980 - 0.333562i
L(12)L(\frac12) ≈\approx 0.433980−0.333562i0.433980 - 0.333562i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.29−0.574i)T 1 + (1.29 - 0.574i)T
3 1 1
5 1+(0.0819−2.23i)T 1 + (0.0819 - 2.23i)T
good7 1+(−1.03−1.03i)T+7iT2 1 + (-1.03 - 1.03i)T + 7iT^{2}
11 1+(3.35+4.61i)T+(−3.39+10.4i)T2 1 + (3.35 + 4.61i)T + (-3.39 + 10.4i)T^{2}
13 1+(0.392+2.48i)T+(−12.3+4.01i)T2 1 + (0.392 + 2.48i)T + (-12.3 + 4.01i)T^{2}
17 1+(4.26+2.17i)T+(9.99+13.7i)T2 1 + (4.26 + 2.17i)T + (9.99 + 13.7i)T^{2}
19 1+(1.72+5.31i)T+(−15.3+11.1i)T2 1 + (1.72 + 5.31i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.0446−0.281i)T+(−21.8−7.10i)T2 1 + (0.0446 - 0.281i)T + (-21.8 - 7.10i)T^{2}
29 1+(−4.78−1.55i)T+(23.4+17.0i)T2 1 + (-4.78 - 1.55i)T + (23.4 + 17.0i)T^{2}
31 1+(−3.26+1.06i)T+(25.0−18.2i)T2 1 + (-3.26 + 1.06i)T + (25.0 - 18.2i)T^{2}
37 1+(3.62−0.573i)T+(35.1−11.4i)T2 1 + (3.62 - 0.573i)T + (35.1 - 11.4i)T^{2}
41 1+(0.181+0.131i)T+(12.6+38.9i)T2 1 + (0.181 + 0.131i)T + (12.6 + 38.9i)T^{2}
43 1+(0.707−0.707i)T−43iT2 1 + (0.707 - 0.707i)T - 43iT^{2}
47 1+(−5.53+2.82i)T+(27.6−38.0i)T2 1 + (-5.53 + 2.82i)T + (27.6 - 38.0i)T^{2}
53 1+(−10.4+5.33i)T+(31.1−42.8i)T2 1 + (-10.4 + 5.33i)T + (31.1 - 42.8i)T^{2}
59 1+(10.3+7.49i)T+(18.2+56.1i)T2 1 + (10.3 + 7.49i)T + (18.2 + 56.1i)T^{2}
61 1+(8.77−6.37i)T+(18.8−58.0i)T2 1 + (8.77 - 6.37i)T + (18.8 - 58.0i)T^{2}
67 1+(0.542−1.06i)T+(−39.3−54.2i)T2 1 + (0.542 - 1.06i)T + (-39.3 - 54.2i)T^{2}
71 1+(11.1+3.62i)T+(57.4+41.7i)T2 1 + (11.1 + 3.62i)T + (57.4 + 41.7i)T^{2}
73 1+(−14.6−2.31i)T+(69.4+22.5i)T2 1 + (-14.6 - 2.31i)T + (69.4 + 22.5i)T^{2}
79 1+(0.564−1.73i)T+(−63.9−46.4i)T2 1 + (0.564 - 1.73i)T + (-63.9 - 46.4i)T^{2}
83 1+(2.71+1.38i)T+(48.7+67.1i)T2 1 + (2.71 + 1.38i)T + (48.7 + 67.1i)T^{2}
89 1+(7.48+10.3i)T+(−27.5+84.6i)T2 1 + (7.48 + 10.3i)T + (-27.5 + 84.6i)T^{2}
97 1+(−2.49−4.90i)T+(−57.0+78.4i)T2 1 + (-2.49 - 4.90i)T + (-57.0 + 78.4i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.03887043438433012248920969545, −8.841106423123243983719115355241, −8.368177655839823117404613582837, −7.44430293289363665746008244514, −6.64845818409723494909796819122, −5.80903284508245664498018554358, −4.91724428471116668549867694589, −3.03404974045309563967262465685, −2.37582243701496533591578347884, −0.34348035800529817020507138980, 1.47112419020502664133575035516, 2.36466032649321911039482796948, 4.11624732244242985137989501628, 4.68944994792576083308603085023, 6.12211750758365481229737678658, 7.22853011924114150224831687261, 7.930157957166459642670264760701, 8.639275701470087896549856316930, 9.448193219805989782446432955571, 10.29040838535992999982353712426

Graph of the ZZ-function along the critical line